Diagnostic method for subsurface hydraulic pumping systems

ABSTRACT

A method is described for determining the downhole operating conditions of a hydraulic pumping system from measurements taken at the surface. Measurements of the surface flow rate and dynamic pressure of power oil used to drive a subsurface pump are made along with determinations of the friction factor of the conduit used to transport the power oil to the pump. This information is then combined to form a boundary valve problem based on the wave equation that describes the acoustic waves in the power oil. Solutions to the wave equation may then be used to determine the downhole conditions.

()1...] 8-7Z OR 3,6359081 lJnlted States Patent [151 3,635,081 Gibbs[451 Jan. 18, 1972 [54] DIAGNOSTIC METHOD FOR Primary Examiner-Jerry W.Myracle SUBSURFACE HYDRAULIC PUMPING Att0rneyJ. H. McCarthy and T. E.Bieber SYSTEMS [57] ABSTRACT [72] Inventor: Sam Gibbs Midland A methodis described for determining the downhole operat- 7 Assign; She 0Company New York ing conditions of a hydraulic pumping system frommeasurements taken at the surface. Measurements of the surface flowFlledl 5, 1970 rate and dynamic pressure of power oil used to drive asubsur- [21] A No; 16,797 f p mp are made along with determinations ofthe friction factor of the conduit used to transport the power oil tothe pump. This information is then combined to form a boundary [52] US.Cl 73/151, 73/168 valve problem based on the wave equation thatdescribes the 1] 2 1 4 /00 acoustic waves in the power oil. Solutions tothe wave equa- [58] Field of Search ..73/l5 l 168 tion may then be usedto determine the downhole conditions. 5 References Cited 4 Claims, 5Drawing Figures UNITED STATES PATENTS 3,354,716 11/1967 Wiebe et al..73/168 X RECORDER 3o PATENTEUmmmz 3.635.081

sum 1 OF 2 3O RECORDER I INVENTOR:

SAM s. GIBBS mzmmmm 3.635081 sum 2 OF 2 s i 8'. I: O 2

I 2 3 TIME,SE(IONDS FIG- 2 a. u] I D m (n LIJ m D.

l 2 3 TIME,SECONDS FIG. 3

R| R2 L2 I C| T C2 FIG- 4 SIGNAL TRANSMISSION ECORDE REC DE R RGENERATOR LINE 3 R FIG. 5

INVENTOR:

SAM G. GIBBS DIAGNOSTIC METHOD FOR SUBSURFACE HYDRAULIC PUMPlNG SYSTEMSBACKGROUND OF THE INVENTION This invention relates to a method fordetemiining the performance characteristics of a pumping well. Moreparticularly it is directed to a method of detennining the downholeconditions of a hydraulic pumping system from surface measurements takenon the power oil used to drive the pumping system.

Subsurface hydraulic pumping systems provide one of the most effectivemeans of lifting large volumes of oil from deep wells, and their use inmodern production operations is rapidly increasing. Typically, asubsurface hydraulic pumping system consists of a reciprocating pumplocated at the production level of a well. Also located at that level isa downhole hydraulic engine for driving the pump. The downhole engine isdriven by high-pressure power oil supplied by a prime mover and pump atthe surface.

Although downhole hydraulic pumping systems have many advantages, theystill lose efficiency over time and occasionally break down. Yet, whenproduction from a well declines, it may be the result of many downholeconditions such as exhaustion of the oil reservoir or a loss in pumpingefficiency. If the loss in production is due to the former, it may beadvisable to shut down the well. But if the only problem is a bad pump,it may be replaced and profitable production resumed. Thus, it is ofcritical importance to know the condition of the downhole pumps. Yet itis very expensive and time consuming to haul them to the surface forvisual inspection. A way of determining downhole pump conditions fromsurface measurements is critically needed.

With sucker rod-type pumping operations, downhole operating conditionscan be determined by an analytical technique utilizing a polished-roddynamometer. The dynamometer is an instrument that records the surfacedata necessary for use in the analytical technique as disclosed in U.S.Pat. No. 3,343,409, Method of Determining Sticker Rod Pump Performance,issued to S. G. Gibbs and copending U.S. Pat. application Ser. No.658,407, filed Aug. 4, 1967, entitled Method and Apparatus for AnalyzingSucker Rod Wave Motion and having the same inventor, now U.S. Pat. No.3,527,094.

However, these techniques are not applicable to subsurface hydraulicpumping systems since there are no sucker rods. Thus, it is a primaryobject of this invention to provide a method of determining downholeoperating conditions for hydraulic pumps from surface measurements.

SUMMARY OF THE INVENTION Broadly, the invention meets the objective ofproviding a method of determining downhole operating conditions in anhydraulically pumped well from data measured at the surface. This isachieved by first determining the friction factor of the conduitdelivering the power oil to the downhole engine. Next the surface levelprime mover and pump are actuated which causes the downhole pump toreciprocate. The dynamic pressure and flow of the power oil is thenmeasured at the surface and recorded as a function of time. Finally, theflow and pressure data are combined with the friction factordeterminations to establish the boundary value problem based on the waveequation that describes the movement of acoustic waves in the power oilas they travel from the downhole pump to the surface. With solutions tothis equation in hand it is possible to accurately determine the dynamicpressure and the flow rate of the power oil at the downhole pump. Thisinformation is in turn used either to construct downhole dynagraph cardsor to determine other downhole factors of interest such as pump intakepressure.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is a schematic diagram,partially in longitudinal section, showing the general arrangement of anhydraulic pumping system for an oil well;

FIG. 2 is a graphical illustration of the surface power oil velocity(flow rate) as a function of time;

FIG. 3 is a graphical illustration of the dynamic pressure of the poweroil measured at the surface;

FIG. 4 is a circuit diagram of an electrical transmission line analogousto the power oil column; and

FIG. 5 is an analog computer utilizing the transmission line of FIG. 4for quickly and accurately determining the downhole power oil pressureand flow rate.

DESCRIPTION OF THE PREFERRED EMBODIMENT Referring now to FIG. 1 there isshown a well having a well casing 10 extending from the reservoir orfluid production interval 12 to the surface 14. Positioned within wellcasing 10 is a production tube 16 having an hydraulic engine 18 and apump 20 located at the production interval 12. A power oil conduit 22interconnects hydraulic engine 18 with a prime mover 24 on the surfacewhich incorporates a pump (not shown). Prime mover and pump 24 supplieshigh-pressure power oil through conduit 22 to run hydraulic engine 18.Pump 20 is a conventional reciprocating pump typically used in downholehydraulic pumping operations. Its function is to lift the oil, water,and/or gas production fluid of reservoir 12 along with the power oil tothe surface through production tube 16 where it is exhausted throughport 26 to storage. In the process, each time pump 20 makes a stroke,pressure and flow rate waves are transmitted to the surface through thepower oil in conduit 22. In effect, the power oil acts as a transmissionline.

The dynamic pressure of the power oil is measured at the surface bypressure transducer 28 and a signal representative thereof istransmitted via electrical lead 29 to recorder 30 where it is recordedon a separate track at some recording speed, V Likewise, the flow rateof power oil is measured by flow rate transducer 32 and a signaltherefrom is transmitted via electrical lead 33 to recorder 30 where itis also recorded on a separate track. The pressure and flow rate arerecorded as a function of time and would look like FIGS. 2 and 3 ifdrawn out on a strip chart.

The pressure and flow rate signals measured by transducers 28 and 30respectively contain information about downhole equipment operation aswell as information about the surface power oil pump 24. The method ofthis invention separates out the information relating to operatingconditions at the downhole engine 18 and pump 20.

As is apparent from FIGS. 2 and 3, the pressure and flow rate signalsare quite complex. In fact the complexity is such that visualinterpretation thereof to infer downhole conditions is difficult, if notimpossible. But if it were possible to measure the dynamic pressure andflow rate of the power oil as it entered the downhole engine, thendownhole indicator cards, similar to dynagraph cards in rod-pumpedwells, could be constructed. Such cards may be interpreted to determinenot only the mechanical conditions and efficiency of the downholeequipment but also to determine the condition of the oil reservoiraround the well bore.

Since measurement of the downhole pressure and flow rate is difficult,it is desirable to obtain such information from surface measurements.Thus, one aspect of this invention is the discovery, through extensiveexperimentation, that the power oil column acts as a good transmissionline and is closely approximated by the following wave equation:

d h(x, t) 2 d h(a:, t) dh(a:, t)

as (w 6 dt (1 where:

a velocity of pressure waves in the power oil in ft./sec. c= dampingfactor r= time in seconds x= distance along the power oil line from thesurface in feet h(x,t) dynamic pressure head g= acceleration of gravityin ft./sec.

Consequently the desired information about dynamic pressure at thedownhole equipment can be determined by solving equation 1 when x isequal to the depth of the downhole equipment.

The damping coefficient is defined by the following equatron:

f( /2 where:

V= average power oil velocity in ft./sec.

f friction factor as determined from a Moody diagram D= internaldiameter of pipe in feet To determine f, it is necessary to calculatethe Reynolds number, the relative roughness of the pipe, and then readthe friction factor from a Moody diagram which is known to those skilledin the art (i.e., at page 3-12 of Handbook of Fluid Mechanics, Streeter,Victor L., McGraw-Hill, 196] Edition).

Reynolds number is given as R=78.64(SG)(V)(D)/p. where:

S0 specific gravity of oil at mean well pressure and temperature V=average power oil velocity in ft./sec.

D internal diameter of pipe in feet viscosity of oil at mean wellpressure and temperature (poise) The relative roughness is given by:

where:

e pipe roughness (inches) D= internal diameter of pipe (inches) Thevelocity of the pressure wave in oil is determined by the followingexpression:

W 1 DC] 1/2 g where:

g acceleration of gravity in ft./sec. D internal diameter of pipe(inches) e thickness of pipe (inches) E= Youngs Modulus for steel 4.32Xl 0"( lb./ft. K= volume modulus of fluid (lb/ft?) W= specific weightoffluid (lb./ft. C 0.95 K is calculated from the compressibility of thefluids in the following manner:

144 (infi/ft?) Compressibility (volum e/volume/psi.)

Thus, the method requires that the above measurements be taken andcombined to determine a and c.

In the solutions of equation 1 a set of boundary conditions isnecessary. These conditions are supplied by the measurements of surfaceflow rate and dynamic pressure of the power oil. Preferably thesemeasurements are in the form of curves of flow rate and dynamic pressureversus time as shown respectively in FIGS. 2 and 3. The boundaryconditions are formulated analytically by approximating these curveswith trun cated Fourier series of the type:

5 60 M0, t) +z v cos not-H sin mot and 11 V(0, i) E 0' cos nwi+rn sinnot where V(0,t) power oil velocity at the surface x=().

The Fourier coefficients 0', 'r, 5, and v of equations 3 and 4 areevaluated from the measured data by conventional har- 7O monic analysis.

Since the mechanics of the solution to equation I with boundaryconditions is not part of this invention, they will not be includedhere. The solutions will merely be stated. However, a detailed solutionto boundary condition problems of this type is set forth in U.S. Pat.No. 3,343,409, Method of Determining Sucker Rod Pump Performance issuedto Gibbs.

The equation for the dynamic pressure head in the power oil line at anydepth x, and at any time t is given by:

1'1 H(1:, l zgPnOv) cos nwH-Qnu) sin Na)! 10 where P(a:) =[(K cosh flx-k6,, sinli fl x) sin u it? t sinh B z+v cosh fl sc) cos a x] (ti) Inthe foregoing equations where T is the period T a',,=gj; V(0, t) cosnwt, n=0, 1, 2,

(a o Bnrn) a2 Kn :w (Braw -mm.)

The above equations 6 and 8 can be solved by programming a digitalcomputer and supplying the test data in the proper form. The exactprogram of the computer can be developed by those skilled in the art,and the details thereof will vary from worker to worker. The computeroutput is a plurality of coordinate points that can be used to plot adownhole card. The selected depth for which the equations are solvedwould be typically that of the downhole pump and engine; however, theequations are general and may be solved for any depth.

An analog computer can also be used to provide a solution to equations 6and 8. An inspection of equation I shows that it has the same form asthe equation of an electrical transmission line: d Q dQ L (Q L W +R a "2w wherein L, R and c inductance, resistance and ca pacitancerespectively, Q charge in coulombs, x distance in feet and t= time inseconds. This equation can be simulated by the circuit of FIG. 4 withmagnitude of the capacitance, resistance and inductance being chosen tocorrespond to the features of the hydraulic system.

The equivalent circuit of FIG. 4 can be incorporated in a special analogcomputer or solved by a general purpose computer. A specialized computeris shown in block diagram form in FIG. 5. The computer of FIG. 5 can beconnected directly to the transducers shown in FIG. 1. The computer willthen supply a dynagraph card for any desired level in the system. Thelevel of the dynagraph card can be selected by scaling the electricalelements of the equivalent transmission line shown in FIG. 4.

The circuit of FIG. 5 uses a recorder 30 to record the pressure and flowrate signals generated by the transducers of FIG. 1. The recorder can bea magnetic tape recorder with provisions for playing back the recordedsignals at approximately I times the recording speed. It is preferredthat the playback speed be greater to reduce the size of the elements inthe equivalent transmission line of FIG. 4 and improve the performanceof the system. The recorder supplies the two signals to a signalgenerator 32 that drives the analog computer to produce a dynagraph forthe desired level.

The transmission line 34 is shown in detail in FIG. 4 and consists ofresistance, inductance and capacitance. The exact size of the elementswill depend upon the physical characteristics of the power oil conduitand its length and the characteristics of the power oil.

The transmission line 34 is connected to the recorder 36 that includesan oscilloscope for displaying the developed signal. Of course, therecorder 36 also includes circuits for adjusting the signals and addingfixed constants such as hydrostatic pressure. The signal displayed onthe oscilloscope will be in the form of a dynagraph card for theselected depth. The signal on the oscilloscope can be photographed toobtain a permanent record of the dynagraph card.

I claim as my invention:

1. A method of determining downhole operating conditions of ahydraulically driven pumping well wherein a reciprocating pump islocated below the fluid level of the well and is driven by a downholehydraulic engine which in turn is driven by high-pressure power oilsupplied through a conduit by a prime mover and pump at the top of thewell, said method comprising:

1. determining the damping coefficient of said power oil conduit bymeasuring the average power oil velocity and physical characteristics ofsaid power oil conduit;

2. determining the pressure wave velocity in the power oil by measuringthe compressibility and specific weight of said power oil and thicknessof said power oil conduit;

3. actuating said prime mover to reciprocate said downhole pump wherebysaid downhole pump generates acoustic waves in said power oil that movefrom said downhole pump to the top of said well; 4. measuring andrecording the dynamic pressure and flow of said power oil as a functionof time;

selecting a depth within the well to be investigated and combining thedamping coefficient determined in step (I) and the pressure wavevelocity determined in step (2) with said dynamic pressure and flow datameasured in step (4) to establish the boundary conditions to the waveequation that describes the movement of acoustic waves in a long columnof high-pressure power oil whereby the dynamic pressure and flow rate ofthe power oil may he determined at said depth; and

plotting a curve of dynamic pressure versus flow rate at said selecteddepth.

2. The method of claim I wherein measuring said damping coefficientcomprises:

measuring the average power oil velocity;

measuring the internal diameter of said power oil conduit;

measuring the roughness of said power oil conduit;

determining the friction factor from a Moody diagram; and

combining the above measurements according to the following expression:

C=f( V/2D) 3. The method of claim 2 wherein determining said pressurewave velocity comprises:

measuring the compressibility of said power oil; measuring the specificweight of said power oil; measuring the thickness of said power oilconduit; and combining the above measurements with the internal diameterof said power oil conduit measured in claim 3 according to the followingequation:

and

1. A method of determining downhole operating conditions of ahydraulically driven pumping well wherein a reciprocating pump islocated below the fluid level of the well and is driven by a downholehydraulic engine which in turn is driven by highpressure power oilsupplied through a conduit by a prime mover and pump at the top of thewell, said method comprising:
 1. determining the damping coefficient ofsaid power oil conduit by measuring the average power oil velocity andphysical characteristics of said power oil conduit;
 2. determining thepressure wave velocity in the power oil by measuring the compressibilityand specific weight of said power oil and thickness of said power oilconduit;
 3. actuating said prime mover to reciprocate said downhole pumpwhereby said downhole Pump generates acoustic waves in said power oilthat move from said downhole pump to the top of said well;
 4. measuringand recording the dynamic pressure and flow of said power oil as afunction of time;
 5. selecting a depth within the well to beinvestigated and combining the damping coefficient determined in step(1) and the pressure wave velocity determined in step (2) with saiddynamic pressure and flow data measured in step (4) to establish theboundary conditions to the wave equation that describes the movement ofacoustic waves in a long column of high-pressure power oil whereby thedynamic pressure and flow rate of the power oil may be determined atsaid depth; and plotting a curve of dynamic pressure versus flow rate atsaid selected depth.
 2. determining the pressure wave velocity in thepower oil by measuring the compressibility and specific weight of saidpower oil and thickness of said power oil conduit;
 2. The method ofclaim 1 wherein measuring said damping coefficient comprises: measuringthe average power oil velocity; measuring the internal diameter of saidpower oil conduit; measuring the roughness of said power oil conduit;determining the friction factor from a Moody diagram; and combining theabove measurements according to the following expression: C f(V/2D) 3.The method of claim 2 wherein determining said pressure wave velocitycomprises: measuring the compressibility of said power oil; measuringthe specific weight of said power oil; measuring the thickness of saidpower oil conduit; and combining the above measurements with theinternal diameter of said power oil conduit measured in claim 3according to the following equation:
 3. actuating said prime mover toreciprocate said downhole pump whereby said downhole Pump generatesacoustic waves in said power oil that move from said downhole pump tothe top of said well;
 4. measuring and recording the dynamic pressureand flow of said power oil as a function of time;
 4. A method ofdetermining the downhole operating conditions of a hydraulically drivenpumping well as set forth in claim 1 wherein said curve is plotted forany desired depth level in the well by utilizing the relationships ofdynamic pressure versus time and flow versus time as expressedrespectively by the following equations:
 5. selecting a depth within thewell to be investigated and combining the damping coefficient determinedin step (1) and the pressure wave velocity determined in step (2) withsaid dynamic pressure and flow data measured in step (4) to establishthe boundary conditions to the wave equation that describes the movementof acoustic waves in a long column of high-pressure power oil wherebythe dynamic pressure and flow rate of the power oil may be determined atsaid depth; and plotting a curve of dynamic pressure versus flow rate atsaid selected depth.